The invention relates to a device and to a method for the controlled achieving of a photon flow between at least one selected resonance of an electromagnetic resonator and a selected target resonance of the resonator, wherein this photon flow in particular supports the redistribution of electromagnetic radiation between the resonances of the resonator in producing a Bose-Einstein photon condensate.
From WO 87/01503 there is known a method and a device for converting electromagnetic waves into monochromatic, coherent electromagnetic radiation with a predeterminable frequency and into heat radiation, wherein the predeterminable frequency lies at the lower edge of the Planck-distributed frequency spectrum of the heat radiation. Thereby, electromagnetic radiation is concentrated in a resonator in a manner such that the average radiation density in the resonator exceeds a critical value and the part of the radiation exceeding this critical value occupies the lowest electromagnetic energy mode of the resonator. The invention WO 87/01503 technically applies the Bose-Einstein condensation in the case of photons.
It is a disadvantage of this known device that the redistribution process for the number of photons exceeding the critical value of the electromagnetic radiation in the resonator is not exactly controllable. If for example the overcritical radiation density with respect to an average radiation temperature is produced by a stationary flow equilibrium, wherein the frequency spectrum of an essential part of the electromagnetic radiation radiated into the resonator lies in the vicinity of a certain resonator resonance, there arises the question of how the photons may flow out of the vicinity of this initial frequency into the fundamental mode of the resonator, wherein the frequency and the number of the net flowing photons adjust to the resonance frequencies which take part. The typical thermalisation processes for electromagnetic radiation in interaction with the resonator walls usually present only a small conversion potential for the Bose-Einstein condensation of photons.
The object of the present invention lies in avoiding the disadvantages of that which is known, in particular to provide a device where photons in the vicinity of a certain frequency, to a greatest extent as possible convert into photons in the region of a predetermined target frequency, wherein the target frequency is smaller than the initial frequency and the photons may be subjected to the modes of a Bose-Einstein condensation so that they may spontaneously flow from the vicinity of the initial frequency into the region of the target frequency which is the fundamental frequency of an electromagnetic resonator.
According to the invention this object is achieved with a device as described below.
Quantum statistical fundamentals
In a stationary flow equilibrium in a photon gas one may create a thermodynamic equilibrium in that the average photon energy density and the average photon number density are fixed independently of one another. This may for example be effected in that with an electromagnetic resonator the wall temperature is fixed whilst with a laser, photons are radiated in. By way of the mutually independent variation of the power and of the frequency of the laser a stationary photon accumulation may be built up whose parametersxe2x80x94average photon number and average photon energyxe2x80x94in pairs, may be set independently of one another. Also the free manipulation of the wall temperature and of the laser power or of the wall temperature and of the laser frequency for this are considered. In place of a laser, also by way of a heat radiation of a suitable temperature which is radiated into the resonator through a long-wave pass filter, there may be created a desired stationary deviation from Plancks""s heat radiation.
Mathematically such a photon gas may be described by way of a so-called xe2x80x9cgrand canonical ensemblexe2x80x9d with an indefinite particle number with the two Lagrange parameters xcex2=1/(kT) and xcexc, with inverse temperature and chemical potential. For the energy density of this free boson gas there applies:                                                                         u                ⁡                                  (                                      β                    ,                    μ                                    )                                            =                                                V                                      -                    1                                                  ⁢                                                      ∑                                                                  k                        =                        1                                            ,                      2                      ,                      …                                                        ⁢                                                                                    ϵ                        k                                            ⁡                                              (                                                                              ⅇ                                                          β                              ⁡                                                              (                                                                                                      ϵ                                    k                                                                    -                                  μ                                                                )                                                                                                              -                          1                                                )                                                                                    -                      1                                                                                                                                              =                                                                    V                                          -                      1                                                        ⁢                                                                                    ϵ                        1                                            ⁡                                              (                                                                              ⅇ                                                          β                              ⁡                                                              (                                                                                                      ϵ                                    1                                                                    -                                  μ                                                                )                                                                                                              -                          1                                                )                                                                                    -                      1                                                                      +                                                      V                                          -                      1                                                        ⁢                                                            ∑                                                                        k                          =                          2                                                ,                        3                        ,                        …                                                              ⁢                                                                                            ϵ                          k                                                ⁡                                                  (                                                                                    ⅇ                                                              β                                ⁡                                                                  (                                                                                                            ϵ                                      k                                                                        -                                    μ                                                                    )                                                                                                                      -                            1                                                    )                                                                                            -                        1                                                                                                                                                    (        1        )            
xcex5k, k=1,2. . . stands for the energy value of the resonator, V for the volume. The second term of the second fine which sums up the energy of all excited modes, for sufficiently xe2x80x9clargexe2x80x9d cavities tends towards                                                         u              e                        ⁡                          (                              β                ,                μ                            )                                =                      6            ⁢                                                            β                                      -                    4                                                  ⁡                                  (                                                                                   xe2x80x2                                        ⁢                    hc                                    )                                                            -                3                                      ⁢                          π                              -                2                                      ⁢                                          g                4                            ⁡                              (                                  ⅇ                                      β                    ⁢                                          xe2x80x83                                        ⁢                    μ                                                  )                                                    ,                  
                ⁢                                            g              α                        ⁡                          (              z              )                                =                                    ∑                                                n                  =                  1                                ,                2                ,                …                                      ⁢                                          z                n                            ⁢                              n                                  -                  α                                                                    ,                            (        2        )            
xe2x80x2h stands for Planck""s constant h divided by 2xcfx80. The parameters xcex2 and xcexc are solutions to the equation system
u(xcex2,xcexc)=u
xcfx81(xcex2,xcexc)=nxe2x80x83xe2x80x83(3)
wherein u indicates the value of the set energy density and n the value of the set photon number. The photon number density xcfx81 as a function of xcex2 and xcexc is given by                                                                         ρ                ⁡                                  (                                      β                    ,                    μ                                    )                                            =                                                V                                      -                    1                                                  ⁢                                                      ∑                                                                  k                        =                        1                                            ,                      2                      ,                      …                                                        ⁢                                                            (                                                                        ⅇ                                                      β                            ⁡                                                          (                                                                                                ϵ                                  k                                                                -                                μ                                                            )                                                                                                      -                        1                                            )                                                              -                      1                                                                                                                                              =                                                                                          V                                              -                        1                                                              ⁡                                          (                                                                        ⅇ                                                      β                            ⁡                                                          (                                                                                                ϵ                                  1                                                                -                                μ                                                            )                                                                                                      -                        1                                            )                                                                            -                    1                                                  +                                                      V                                          -                      1                                                        ⁢                                                            ∑                                                                        k                          =                          2                                                ,                        3                        ,                        …                                                              ⁢                                                                  (                                                                              ⅇ                                                          β                              ⁡                                                              (                                                                                                      ϵ                                    k                                                                    -                                  μ                                                                )                                                                                                              -                          1                                                )                                                                    -                        1                                                                                                                                                    (        4        )            
For sufficiently large cavities the second term in (4), the term of the excited modes results in
xcfx81e(xcex2,xcexc)=2xcex2xe2x88x923(xe2x80x2hc)xe2x88x923xcfx80xe2x88x922g3(excex2xcexc)xe2x80x83xe2x80x83(5)
For the chemical potential there applies
xcexcxe2x89xa6xcex51xe2x80x83xe2x80x83(6)
so that the occupation probabilities occuring in (4) may not become negative.
With an increasing size of the cavity, xcex51 reciprocally to the characteristic xe2x80x9cdiameterxe2x80x9d tends to 0. In the limit case of infinitely large cavities xcexc is negative or equal to 0. If xcexc equal to 0 excited modes absorb the maximum energy density
uc(xcex2):=ue(xcex2,0)xe2x80x83xe2x80x83(7)
This is the energy density of the black body radiation. If there is set an energy density u which exceeds this value, the energy excess must be taken up by the fundamental mode, the first term in (1). In the ideal case of an infinitely large cavity, i.e. for each sufficiently large cavity, in a good approximation there then applies
Vxe2x88x921xcex51(excex2(xcex51xe2x88x92xcexc)xe2x88x921)xe2x88x921=uxe2x88x92u c(xcex2)xe2x80x83xe2x80x83(8)
Expanding the exponential function on the left side in
Vxe2x88x921xcex51(xcex51xe2x88x92xcexc)xe2x88x921xcex2xe2x88x921xe2x80x83xe2x80x83(9)
then it is obvious that xcexcxe2x88x92xcex51 reciprocally to the fourth power of the characteristic diameter of the cavity tends to 0.
If the fundamental mode energy is different to 0, which means that, the fundamental mode is occupied macroscopically, the photon number in the fundamental mode, thus the first term in (4), becomes singular in that it increases proportionally to the diameter of the cavity. This is plausible since an infinitely large number of photons of infinitesimal energy gives a finite energy term. This is precisely the infrared singularity.
On the Mechanism of the Redistribution of Photons
Bose-Einstein condensation of photons means that the photons exceeding the critical energy density uc(xcex2) transfer into the fundamental mode of the resonator. This is possible by the interaction of the photons with the wall of the resonator. Since the quality factor of the cavity has a finite value a broadening of the resonances and thus an overlapping of the resonance curves result. This implies non-zero transition probabilities between the resonances. A cavity which suits for photon condensation may be designed such that the transition probabilities dominate the absorption probabilities.
Ignoring the Frequency Shift
ƒk(t)xe2x88x9deixcfx89ktxe2x88x92xcfx89kt/(2Qk)xe2x80x83xe2x80x83(10)
is the photon function in the resonance k taking into account the damping. The quality factor Q is, up to a geometric factor of the order 1, given by (see John David Jackson, Classical Electrodynamics, Second Edition, John Wiley and Sons, New York, 1975, p. 359)
Qk=(xcexc0"sgr"xcfx89k/2)xc2xdV/Axe2x80x83xe2x80x83(11)
"sgr" is the conductivity of the wall material, xcexc0 the magnetic permeability of the vacuum, V the volume and A the surface area of the cavity. The Fourier transform of (10) is                                                                         f                k                            ⁡                              (                ω                )                                      ∝                                          (                                                      -                                          ⅈ                      ⁡                                              (                                                  ω                          -                                                      ω                            k                                                                          )                                                                              -                                                            ω                      k                                        /                                          (                                              2                        ⁢                                                  Q                          k                                                                    )                                                                      )                                            -                1                                              =                                    (                                                -                                      ⅈ                    ⁡                                          (                                              ω                        -                                                  ω                          k                                                                    )                                                                      -                                  α                  ⁢                                      xe2x80x83                                    ⁢                                      ω                                          1                      /                      2                                                                                  )                                      -              1                                      ,                  
                ⁢                  α          =                                                    (                                  2                  ⁢                                      μ                    0                                    ⁢                  σ                                )                                                              -                  1                                /                2                                      ⁢                          A              /              V                                                          (        12        )            
The probability amplitude for a transition xcfx89kxe2x86x92xcfx891 is                               Ta          ⁡                      (                                          ω                k                            ,                              ω                l                            ,              α                        )                          =                              ∫            0            ∞                    ⁢                                                    f                k                            ⁡                              (                ω                )                                      ⁢                                          f                l                cc                            ⁡                              (                ω                )                                      ⁢                          xe2x80x83                        ⁢                          ⅆ                                                ω                  ⁡                                      (                                                                  ∫                        0                        ∞                                            ⁢                                                                                                    f                            k                                                    ⁡                                                      (                            ω                            )                                                                          ⁢                                                                              f                            k                            cc                                                    ⁡                                                      (                            ω                            )                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                  ⅆ                          ω                                                                                      )                                                                                        -                    1                                    /                  2                                                      ⁢                                          (                                                      ∫                    0                    ∞                                    ⁢                                                                                    f                        l                                            ⁡                                              (                        ω                        )                                                              ⁢                                                                  f                        l                        cc                                            ⁡                                              (                        ω                        )                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      ω                                                                      )                                                              -                  1                                /                2                                                                        (        13        )            
fcc denotes the conjugate complex of the function f. The transition probability results from the multiplication of the probabaility amplitude by its conjugate complex to
Tp(xcfx89k,xcfx89l,xcex1)={(xcfx80+Arc Tan[xcfx89
kxc2xd/xcex1]+Arc Tan[xcfx89lxc2xd/xcex1])
2+4xe2x88x921Log[(xcfx89kxcex12+xcfx89k2)/(xcfx89lxcex1
2+xcfx89l2)]}**(Arc Tan[xcfx89kxc2xd/xcex1]+xcfx80/2)
xe2x88x921(Arc Tan[xcfx89lxc2xd/xcex1]+xcfx80/2)xe2x88x921**((xcfx89
kxc2xd+xcfx89lxc2xd)2xcfx89kxe2x88x92xc2xdxcfx89
lxe2x88x92xc2xd+(xcfx89kxe2x88x92xcfx89l)2xcex1xe2x88x922xcfx89k
xe2x88x92{fraction (1/2)}xcfx89lxe2x88x92xc2xd)xe2x88x921xe2x80x83xe2x80x83(14)
For frequencies close to one another, i.e. if x=xcfx89kxe2x88x92xcfx89l is small, then approximately
Tp[xcfx89,xcfx89+x,xcex1]≈{4xe2x88x92(x/xcfx89)(xcex12+2xcfx89+x)/(4xcfx80
2(xcex12+xcfx89))}**(4+x2xcfx89xe2x88x921xcex1xe2x88x922)xe2x88x921xe2x80x83xe2x80x83(15)
This transition probability is close to 1 if
x2xcfx89xe2x88x921xcex1xe2x88x922 less than 1xe2x80x83xe2x80x83(16)
This may be exploited as a design criterium for a resonator for the photon condensation.
Example of cuboidal cavity: b less than a less than 1
(see e.g. Peter A. Rizzi, Microwave Engineeringxe2x80x94Passive Circuits,. Prentice Hall, Engelwood Cliffs, N.J., 1988)
vm,n,p=xcfx89m,n,p/(2xcfx80)=c2xe2x88x921((m/a)
2+(n/b)2+(p/l)2)xc2xdxe2x80x83xe2x80x83(17)
The lowest energy inherent eigen value is
xe2x80x83v1,0,1=(c/2)(axe2x88x922+1xe2x88x922)xc2xdxe2x80x83xe2x80x83(18)
and the difference between two neighbouring resonances in the lower frequency region where their maximum is to be expected is                                                         x              =                                                ω                                      1                    ,                    0                    ,                                          p                      +                      1                                                                      -                                  ω                                      1                    ,                    0                    ,                    p                                                                                                                          =                              π                ⁢                                  xe2x80x83                                ⁢                c                ⁢                                  {                                                                                    (                                                                              a                                                          -                              2                                                                                +                                                                                    (                                                                                                (                                                                      p                                    +                                    1                                                                    )                                                                /                                l                                                            )                                                        2                                                                          )                                                                    1                        /                        2                                                              -                                                                  (                                                                              a                                                          -                              2                                                                                +                                                                                    (                                                              p                                /                                l                                                            )                                                        2                                                                          )                                                                    1                        /                        2                                                                              }                                                                                                        =                                                π                  ⁡                                      (                                          c                      /                      a                                        )                                                  ⁢                                  2                                      -                    1                                                  ⁢                                  (                                                            2                      ⁢                      p                                        +                    1                                    )                                ⁢                                                      (                                          a                      /                      l                                        )                                    2                                ⁢                                                      (                                          1                      +                                                                                                    p                            2                                                    ⁡                                                      (                                                          a                              /                              l                                                        )                                                                          2                                                              )                                                                              -                      1                                        /                    2                                                                                                          (        19        )            
For calculating the maximum value of x2/xcfx89 in (16) the maximum value of the factor
(2p+1)2(1+p2(a/l)2)xe2x88x92{fraction (3/2)}
is computed which is reached for
p=xe2x88x92xc2xe+({fraction (9/16)}+2(l/a)2)xc2xd≈2xc2xdl/axe2x80x83xe2x80x83(20)
For the cuboid there thus results the design criterium (16) approximately
x2xcfx89xe2x88x921xcex1xe2x88x922≈xcfx80cxcexc0"sgr"*2*3xe2x88x92{fraction (3/2)}
*a*lxe2x88x922*(lxe2x88x921+axe2x88x921+bxe2x88x921)xe2x88x922xe2x80x83xe2x80x83(21)
In the case that b is small with respect to a and l there results in SI units for (21) the approximation
228"sgr"ab2lxe2x88x922xe2x80x83xe2x80x83(22)
Example of a circular cylinder: (Jackson p.356)
R: circular radius
d: height
xcex5: dielectricity constant
xcexc: magnetic permeability
xe2x80x83xcfx89m,n,p=(xcexcxcex5)xe2x88x92xc2xd(xxe2x80x2mn2Rxe2x88x922+p2xcfx802dxe2x88x922)xc2xd=(xcexcxcex5)xe2x88x92xc2xdxxe2x80x2mnRxe2x88x921(1+R2p2xcfx802/(dxxe2x80x2mn)2)xc2xdxe2x80x83xe2x80x83(23)
xxe2x80x2mn, the root of J |m(x) assumes the following values:
For estimating the maximal distance of two resonances which accordingly entails the most unfavourable transition probability we approximatively compute the difference according to the series expansion of the root and obtain                                                                                           ω                                      1                    ,                    1                    ,                                          p                      +                      1                                                                      -                                  ω                                      1                    ,                    1                    ,                    p                                                              =                              xe2x80x83                            ⁢                                                cx                                      1                    ,                    1                                    xe2x80x2                                ⁢                                  R                                      -                    1                                                  ⁢                                  2                                      -                    1                                                  ⁢                                  (                                                            2                      ⁢                      p                                        +                    1                                    )                                ⁢                                  R                  2                                ⁢                                                      π                    2                                    /                                                            (                                                                        dx                          ⁢                                                      xe2x80x83                                                                                                    1                          ,                          1                                                xe2x80x2                                            )                                                              2                      ⁢                                              xe2x80x83                                                                                                                                                                                    xe2x80x83                            ⁢                                                (                                      1                    +                                                                  R                        2                                            ⁢                                              p                        2                                            ⁢                                                                        π                          2                                                /                                                                              (                                                          dx                              11                              xe2x80x2                                                        )                                                    2                                                                                                      )                                                                      -                    1                                    /                  2                                                                                                        =                              xe2x80x83                            ⁢                                                (                                                            2                      ⁢                      p                                        +                    1                                    )                                ⁢                                                      (                                          1                      +                                                                        R                          2                                                ⁢                                                  p                          2                                                ⁢                                                                              π                            2                                                    /                                                                                    (                                                              dx                                11                                xe2x80x2                                                            )                                                        2                                                                                                                )                                                                              -                      1                                        /                    2                                                                                                                                          xe2x80x83                            ⁢                                                2                                      -                    1                                                  ⁢                c                ⁢                                  xe2x80x83                                ⁢                                  π                  2                                ⁢                                  x                  11                                      xe2x80x2                    ⁢                                          xe2x80x83                                        -                    1                                                  ⁢                                  Rd                                      -                    2                                                                                                          (        24        )            
With x=xcfx891,1, p+1xe2x88x92xcfx891,1,p one calculates the left side in the design criterium (16). In order to estimate the most unfavourable case one determines the maximum of
(2p+1)2(1+R2p2xcfx802/(dxxe2x80x211)2)xe2x88x92{fraction (3/2)}
for
p=xe2x88x92(xc2xe)+(({fraction (9/16)})+2(xxe2x80x211/xcfx80)2(d/R)2)xc2xd≈2xc2xd(xxe2x80x211/xcfx80)(d/R)
wherein the last approximation applies when the cylinder height d is large compared to the circular diameter R. For the circular cylinder the design criterium (16) thus results in
x2xcfx89xe2x88x921xcex1xe2x88x922≈xcfx8023xe2x88x923/2 xxe2x80x211xe2x88x921cxcexc0"sgr"Rdxe2x88x922(dxe2x88x921+Rxe2x88x921)xe2x88x922≈389"sgr"R3dxe2x88x922xe2x80x83xe2x80x83(25)
the last approximation is again to be understood in SI units.
Absorption in Competition With the Photon Transitions Between the Resonances
A mechanism for the net redistribution of photons of higher frequencies into photons of lower frequencies as is necessary for a Bose-Einstein condensation of photons is possible when the transition probabilities between the resonances are always greater than the absorption probabilities. According to (12) with the normalisation factor N, the form function of the resonance k results in
|ƒk(xcfx89)|2=Nxe2x88x922((xcfx89xe2x88x92xcfx89k)2+xcfx89k2(2Qk)xe2x88x922)xe2x88x921xe2x80x83xe2x80x83(26)
The form function (26) for xcfx89k assumes the maximum value
|ƒl(xcfx89k)|2=Nxe2x88x922xcfx89kxe2x88x922(2Qk)2xe2x80x83xe2x80x83(27)
The half width follows from
|ƒk(xcfx89)|2=2xe2x88x921Nxe2x88x922xcfx89kxe2x88x922(2Qk)2xe2x80x83xe2x80x83(28)
to
2(xcfx89xe2x88x92xcfx89k)=xcfx89k/Qkxe2x80x83xe2x80x83(29)
The decay time Qk/xcfx89k substituted into the photon function (10) indicates how long it lasts until the resonance decays to the e. part. I.e. after
Qk/xcfx89kxe2x80x83xe2x80x83(30)
seconds the absorption probability of the photons of a resonance is
lxe2x88x92exe2x88x921=63.2%xe2x80x83xe2x80x83(31)
The decay time (30) simultaneously gives the time scale for photon transitions between the resonances.
Criterium: If all transition probabilities between neighbouring resonances of a resonator are larger than (31) there results a redistribution excess.
According to the invention the device serves the controlled achieving of a photon flow between a selected resonance of an electromagnetic resonator and a selected target resonance of the resonator, wherein this photon flow in particular supports the redistribution of electromagnetic radiation between the resonances of the resonator with the production of a Bose-Einstein photon condensate. It consists essentially of a cavity with reflecting walls and of means for coupling electromagnetic radiation into the cavity, wherein the means are designed in a manner such that the average energy density of the electromagnetic radiation reaches a value which is larger than the critical energy density
ucrit(xcex2):=ue(xcex2,0)
at the average, effectively set temperature T of the radiation in the resonator.
With this xcex2=1/(kT), k is the Boltzmann constant,                     u        e            ⁡              (                  β          ,          μ                )              =          6      ⁢                                    β                          -              4                                ⁡                      (                                                           xe2x80x2                            ⁢              hc                        )                                    -          3                    ⁢              π                  -          2                    ⁢              g        4            ⁢              (                  ⅇ                      β            ⁢                          xe2x80x83                        ⁢            μ                          )              ,      
    ⁢                    g        α            ⁡              (        z        )              =                  ∑                              n            =            1                    ,          2          ,          …                    ⁢                        z          n                ⁢                  n                      -            α                                ,
xe2x80x2h the Planck""s constant divided by 2xcfx80. The parameters xcex2 and xcexc are determined by the solution of the equation system
u(xcex2,xcexc)=u
xcfx81(xcex2,xcexc)=n
where u denotes the value of the set energy density and n the value of the photon number density which set is (see e.g. Res Jost, Quantenmechanik II, Verlag der Fachvereine der ETZH Zxc3xcrich, 1973, p. 151 ff.). Via this equation system the variables u and n are implicitly related to the temperature. For example it is technically comfortable to observe the average energy density u and the effective temperature T of the radiation as independent variables.
The coupled-in electromagnetic waves initially occupy the cavity modes with the respective frequencies.
According to the invention the device is designed such that the transition probability for the transition of photons between neighbouring modes in the range between the initial resonance frequencies and the target resonance frequency is larger than the probability for the absorption of photons. In particular this may be achieved by the selection of the reflectivity of the walls of the cavity, the shape of the cavity, the size of the cavity or also by way of a medium incorporated into the cavity.
The advantage of this device lies in the fact that the redistribution procedure of photons via the resonance modes of a cavity may be controlled and direct influenced and no longer remains dependent on accidental thermalisation processes. This device thus simplifies the Bose-Einstein condensation of photons. In its application as a solar cell, as this is described in WO 87/01503, with the redistribution device for photons described here also the efficiency for producing the useful, laser-like ground state may be controlled and in particular increased in a targeted manner.